Convergence analysis of a weak Galerkin finite element method on a Shishkin mesh for a singularly perturbed fourth-order problem in 2D

We consider the singularly perturbed fourth-order boundary value problem $\varepsilon ^{2}\Delta ^{2}u-\Delta u=f$ on the unit square $\Omega \subset \mathbb{R}^2$, with boundary conditions $u = \partial u / \partial n = 0$ on $\partial \Omega$, where $\varepsilon \in (0, 1)$ is a small parameter. The problem is solved numerically by means of a weak Galerkin(WG) finite element method, which is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on finite element partitions consisting of polygons of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Under reasonable assumptions on the structure of the boundary layers that appear in the solution, a family of suitable Shishkin meshes with $N^2$ elements is constructed ,convergence of the method is proved in a discrete $H^2$ norm for the corresponding WG finite element solutions and numerical results are presented

Анализ влияния исламского возрождения в государствах Центральной Азии на безопасность в Синьцзяне Китая

After the Arabs invaded Central Asia, Islam began to dominate the region. Although during the Soviet Union due to socialist ideology Islam in the region was diluted, after the independence of the five Central Asian countries, Islam still met with the revival. Especially in the 1980s, some Islamic groups in Central Asia began to pay close attention to the Islamic revolution in Iran and the jihad in Afghanistan, the ideas of radicalism under Islamic slogans also began to penetrate. In fact, during the Soviet era, Islam was still generally considered by the people as a way of moral recovery of society. In the mind, morality and Islam have almost always been identified. This article examines the resurgence of Islam in Central Asia after the collapse of the Soviet Union and the spread of extremist Islamic ideas in the region, analyzes the impact of the Islamic renaissance in Central Asia on Xinjiang China, especially the influence of the extremist organization Hizb ut-Tahrir on the activities of Chinese "three forces of evil" and cross-border crimes, arms smuggling and drug trafficking. The author believes that although the Islamic revival movement in different countries had a great influence on the development of Islamic fundamentalism (IF) and even extremist ideology in China, the Islamic revival movement in Central Asia had a more direct impact on the security situation in Xinjiang. Therefore, the Xinjiang local government in recent years has strengthened control and surveillance at the border of China and Central Asia and is carrying out a number of measures to control the area, aimed at preventing the spread of extreme ideas and terrorist activities. Although now the main part of the IF was defeated, judging by the prevailing situation, the security situation in the Fergana region of Central Asia, which is located on the border of Uzbekistan, Kyrgyzstan and Tajikistan, is still the most serious in the future. The development of remnants and branches in Central Asia and their threat to China's security are still underestimated.После того, как арабы вторглись в Центральную Азию, ислам начал доминировать в этом регионе. Во время СССР из-за социалистической идеологии ислам в регионе не имел условий для развития. После получения независимости пяти центрально-азиатских стран, началось возрождение ислама. Особенно в 1980-е годы некоторые исламские группы в ЦА уделяли пристальное внимание исламской революции в Иране и джихаду в Афганистане, куда начали проникать идеи радикализма под исламскими лозунгами. В данной статье рассматривается возрождение ислама в Центральной Азии после распада Советского Союза и распространение экстремистских исламских идей в регионе, проводится анализ влияния исламского возрождения в Центральной Азии на Синьцзян в Китае. Прежде всего, влияния экстремистской организации «Хизб ут-Тахрир» на деятельность «трех силы зла» и трансграничные преступления, контрабанду оружия и торговлю наркотиками. Хотя движение исламского возрождения в разных странах оказало большое влияние на развитие исламского фундаментализма и даже экстремистской идеологии в Китае, движение исламского возрождения в Центральной Азии сказалось прежде всего, на обстановке в Синьцзяне. Поэтому синьцзянское местное правительство в последние годы усилило контроль и наблюдение на границе Китая и Центральной Азии и проводит ряд мероприятий, направленных на предотвращение распространения экстремальных идей и террористической деятельности. Хотя сейчас основную часть незаконных организаций разгромили, в будущем серьезной проблемой по-прежнему является ситуация в Ферганском регионе ЦА, который расположен на границе Узбекистана, Кыргызстана и Таджикистана

Global dynamics for a class of reaction–diffusion multigroup SIR epidemic models with time fractional-order derivatives

This paper investigates the global dynamics for a class of multigroup SIR epidemic model with time fractional-order derivatives and reaction–diffusion. The fractional order considered in this paper is in (0; 1], which the propagation speed of this process is slower than Brownian motion leading to anomalous subdiffusion. Furthermore, the generalized incidence function is considered so that the data itself can flexibly determine the functional form of incidence rates in practice. Firstly, the existence, nonnegativity, and ultimate boundedness of the solution for the proposed system are studied. Moreover, the basic reproduction number R0 is calculated and shown as a threshold: the disease-free equilibrium point of the proposed system is globally asymptotically stable when R0 ≤ 1, while when R0 > 1, the proposed system is uniformly persistent, and the endemic equilibrium point is globally asymptotically stable. Finally, the theoretical results are verified by numerical simulation

Additive Modulation of DNA-DNA Interactions by Interstitial Ions

Quantitative understanding of biomolecular electrostatics, particularly involving multivalent ions and highly charged surfaces, remains lacking. Ion-modulated interactions between nucleic acids provide a model system in which electrostatics plays a dominant role. Using ordered DNA arrays neutralized by spherical cobalt3+ hexammine and Mg2+ ions, we investigate how the interstitial ions modulate DNA-DNA interactions. Using methods of ion counting, osmotic stress, and x-ray diffraction, we systematically determine thermodynamic quantities, including ion chemical potentials, ion partition, DNA osmotic pressure and force, and DNA-DNA spacing. Analyses of the multidimensional data provide quantitative insights into their interdependencies. The key finding of this study is that DNA-DNA forces are observed to linearly depend on the partition of interstitial ions, suggesting the dominant role of ion-DNA coupling. Further implications are discussed in light of physical theories of electrostatic interactions and like-charge attraction